Projections Maps, Shellings and Duality

PROJECTIONS, SHELLINGS AND DUALITY

SWAPNEEL MAHAJAN

Abstract. Projection maps which appear in the theory of buildings and oriented matroids are closely related to the notion of shellability. This was first observed by Björner [9]. In this paper, we give an axiomatic treatment of either concept and show their equivalence. We also axiomatize duality in this setting. As applications of these ideas, we prove a duality theorem on buildings and give a geometric interpretation of the flag h vector. The former may be regarded as a q-analogue of the Dehn-Sommerville equations. We also briefly discuss the connection with the random walks introduced by Bidigare, Hanlon and Rockmore [5].

Contents 1. Introduction 1 2. Projection maps 2 3. Shellings and restriction maps 5 4. Axioms 8 5. An almost example 13 6. The metric setup 15 7. Type selected subcomplexes 17 8. Commutativity issues 21 9. The opposite or duality axiom 24 10. Duality in Buildings 27 11. Future prospects 30 References 32

1. Introduction Projection maps appeared independently in the theory of buildings developed by Tits [33] and in the theory of oriented matroids [10]. Given the importance of these maps in either theory, it is reasonable to try to formulate them axiomatically. The close connection between these maps and the notion of shellability was ﬁrst observed by Bj¨orner [9]. In this paper, we study projection maps axiomatically keeping the viewpoint of [9]. This leads us to the notion of many “compatible” shellings rather than a single shelling. To complete the picture, we also consider restriction maps that are useful to keep track of a shelling. For ideas closely related to this paper, see [27, 16, 21]. The main result of this paper (Theorem 1) may be

1 2000 Mathematics Subject Classification. Primary 06A08, 52B22, 52B30. Key words and phrases. projection map; shelling; left regular band; h vector; random walk; building; duality. 1 2 SWAPNEEL MAHAJAN informally stated as Projection maps ⇐⇒ Compatible shellings. Thus far, people seem to be interested in the question “Is a complex shellable?” The next question that should be asked is “How do we study the space of all shellings of a complex?” As explained in the previous paragraph, projection maps give us an approach to this hard problem. We mention that though compatible shellings form a fairly large class of shellings, they are far from giving all possible shellings of a complex. For a labeled shellable complex ∆, Björner [9, Theorem 1.6] gave a simple geometric interpretation of the flag h vector. In our case, ∆ has many “compatible” shellings rather than a single shelling. This allows us to define local flag h vectors of ∆. And the average of the local vectors gives us the usual vector. Also the local vectors have an even simpler geometric interpretation. Further the local vectors are all equal to the usual vector when the projection maps satisfy some commutativity relations. These relations are related to the uniformity of the stationary distribution of certain random walks introduced by Bidigare, Hanlon and Rockmore [5]. In this context, we generalize a result on uniform stationary distributions obtained in [6]. Next we formalize the notion of duality by adding an “opposite” axiom to our axiomatic setup. It is strong enough to imply the Dehn-Sommerville equations. And as one expects, this axiom can hold only for a simplicial complex homotopy equivalent to a sphere. From the viewpoint of shellings, the relevant concept is that of shelling reversal, i.e., when is the reverse of a shelling again a shelling? The Solomon-Tits theorem says that a (thick) spherical building ∆ has the homotopy type of a (non-trivial) wedge of spheres. Hence by the observation in the previous paragraph, ∆ cannot satisfy the opposite axiom. However ∆ has a re- markable duality which can be expressed in terms of the flag h vector (Theorem 6). This may be regarded as a q-analogue of the Dehn-Sommerville equations.

Organization of the paper. The next two sections provide basic definitions and motivating examples. The axiomatic theory involving projection maps, shellings and restriction maps is presented in Section 4 and later in a more intuitive metric setup in Section 6. In Section 7 we explain the connection with the flag h vector. The connection with random walks is explained in Section 8. In the next two sections, we study the notion of duality first for thin complexes and then for buildings. In the final section, we outline some problems for further study.

2. Projection maps In this section, we give four examples of the mini-theory that we will present in Section 4. We will return to these examples again in Section 6, where we will show that they are indeed examples of our theory. Terms like gallery connected, convex, etc. that we freely use here are also explained in that section. For now, we present the examples from the viewpoint of projection maps. They ﬁt into the framework of LRBs (non-associative in general) as indicated in Table 1.

The question marks say that a building-like analogue for the more general case of hyperplane arrangements is unknown. Though LRBs in general are not examples PROJECTIONS, SHELLINGS AND DUALITY 3

Table 1. Examples of the theory.

LRB Non-associative LRB Hyperplane arrangements ??? Reﬂection arrangements Buildings Distributive lattices Modular lattices of our theory they come quite close as we will see in Section 5; and hence can potentially give more examples. Keeping this in mind, we start with a brief review of LRBs. Our exposition in Section 2.1 is mainly taken from [14].

2.1. Left regular bands. Let S be a semigroup (finite, with identity). A left- regular band, or LRB, is a semigroup S that satisfies the identities (D) x2 = x and xyx = xy for all x,y ∈ S. We call (D) the “deletion property”, because it admits the following restatement: Whenever we have a product x1x2 ··· xn in S, we can delete any factor that has occurred earlier without changing the value of the product. Alternatively, one can say that S is a LRB if there are a lattice L and a surjection supp: S ։ L satisfying (1) supp xy = supp x ∨ supp y and (2) xy = x if supp y ≤ supp x. Here ∨ denotes the join operation (least upper bound) in L. A good reference for LRBs is [14]. It explains the equivalence of the above two definitions and also contains plenty of examples. More information about LRBs can be found in [17, 24, 25]. Early references to the identity xyx = xy are [19, 29]. We can also define a partial order on a LRB by setting (3) x ≤ y ⇐⇒ xy = y. We now switch to a slightly different notation which we will be using for the most part. We denote a LRB by F and call elements of F faces. A face C is a chamber if CF = C for all F ∈ F, or equivalently, if it is maximal in the partial order on F specified by equation (3). The set of chambers, which we denote C, is an ideal in F. This gives us a map F×C→C that maps (F,C) to FC. We call FC the projection of C on F . In this paper, we are mainly interested in these projection maps rather than the full product in F. Example 1. Hyperplane arrangements: The motivating example of a LRB is the poset of regions of a central hyperplane arrangement (or more generally of an oriented matroid). A good reference for this example is [14, Appendix A]. More details can be found in [5, 6, 10, 13, 15, 23, 34]. Briefly, a finite set of linear hyperplanes (i.e. hyperplanes passing through the origin) in a real vector space V divides V into regions called chambers. These are polyhedral sets, which have faces. The totality F of all the faces is a poset under the face relation. Less obviously, F admits a product, making it a LRB. The lattice L is the intersection lattice of the arrangement (or more generally the underlying matroid). 4 SWAPNEEL MAHAJAN

The product can be described combinatorially by encoding faces using sign sequences. In fact one way to axiomatize an oriented matroid is in terms of sign sequences and this product. The deletion property (D) for F can be checked directly. The product in F can also be described geometrically. For instance, the projection FC is the chamber closest to C having F as a face. The case that is directly relevant to our theory is that of simplicial hyperplane arrangements, i.e., the chambers are simplicial cones and hence F is a simplicial complex. As an interesting example, we mention Coxeter complexes which arise from reflection arrangements. Now let D⊆C be a convex set of chambers and let G be the set of faces of all the chambers in D. Then using the geometric description of the product in F, for instance, one can check that G is also a LRB. We now give a concrete example of this type. Example 2. Distributive lattices: A good reference for this example is [14, Section 4]. We will soon see that it generalizes to the case of modular lattices (Example 4). We start with the basic example of the Boolean lattice Bn of rank n consisting of all subsets of an n-set ordered under inclusion. Its flag (order) complex ∆(Bn) is the Coxeter complex of type An−1 and corresponds to the braid arrangement. Hence the set of faces of ∆(Bn) is a LRB. Moreover the product can be described entirely using the lattice structure (meets and joins) of Bn. More generally, the set of faces F of the flag (order) complex ∆(M) of any distributive lattice M is also a LRB. A simple way to verify this is to appeal to the well-known fact that M can be embedded as a sublattice of the Boolean lattice. More geometrically, Abels [1, Proposition 2.5] has described a way of constructing an embedding which makes the set of chambers in ∆(M) (i.e., the maximal chains in M) correspond to a convex set of chambers in ∆(Bn), the complex of the braid arrangement. 2.2. Non-associative LRBs. As one may expect, these are the same as LRBs except that we no longer require associativity. However, we do require that xyx be well-defined for all x,y; that is, xyx = (xy)x = x(yx). With this restriction, the deletion property makes sense. In the non-associative setting, the second definition of LRBs involving a lattice L does not make much sense and we omit it. Also transitivity of the relation ≤ defined by equation (3) is no longer automatic. So we have two choices; either to take transitive closure of the relation or simply to impose transitivity as an additional condition. In the two examples that we consider, transitivity of the relation ≤ is in fact automatic and hence we do not pursue this issue further. To avoid confusion later, we mention that the term LRB always means an associative LRB. Whenever we want to include the non-associative case, we will say so explicitly. Example 3. Buildings: A good reference for this example is [33]; also see [13, 28]. Some of the terminology used here is explained at the beginning of Section 6. Also the building of type An−1 is described briefly in Example 5 in Section 10. Let W be a Coxeter group and Σ(W ) its Coxeter complex. Roughly a building ∆ of type W is a union of subcomplexes Σ (called apartments) which fit together nicely. Each apartment Σ is isomorphic to Σ(W ). For any two simplices in ∆, there is an apartment Σ containing both of them. As a simplicial complex, ∆ is pure, PROJECTIONS, SHELLINGS AND DUALITY 5 labeled and gallery connected. Also each apartment is convex. There is a W -valued distance function δ : C × C → W that generalizes the gallery distance. Furthermore for any apartment Σ and chamber C ∈ C, there is a retraction ρ = ρΣ,C : ∆ → Σ satisfying δ(C, ρ(D)) = δ(C,D) for any D ∈C. We denote the set of faces by F. For F, G ∈ F, we choose an apartment Σ containing F and G and define F G to be their product in Σ. Since Σ is a Coxeter complex, we know how to do this (Example 1). Furthermore it can be shown that the product does not depend on the choice of Σ. Hence this defines a product on the set of faces F of ∆. And it is compatible with the retraction ρ = ρΣ,C , namely, ρ(F )C = ρ(FC) for any face F of ∆. Example 4. Modular lattices: A good reference for this example is [1] where you can find proofs of all the facts that we state here. The flag (order) complex ∆(M) of a modular lattice M is a labeled simplicial complex. A face of ∆ is a chain in M. As usual, we denote the set of faces by F. The chambers C of ∆ are the maximal chains in M. We define the map F×C→C as follows. For F ∈F and C ∈C, define FC to be the unique chamber containing F that is contained in the sublattice generated by F and C. More generally, for F, G ∈F, the face F G is the chain in M obtained by refining the chain F by the chain G, using meets and joins as in a Jordan–Hölder product. An interesting subclass of modular lattices is that of distributive lattices that we discussed earlier. Another interesting fact is that in a modular lattice, the sublattice generated by any two chains is distributive; see [7, pg 66]. Hence modular lattices may be regarded as generalizations of buildings of type An−1, with the distributive lattices playing the role of apartments (and their convex subsets). It may be possible to generalize this example in various directions. For example, one may consider more general lattices like supersolvable lattices [31]. It is also a challenge to find analogues of modular lattices that generalize buildings of types other than An−1. Remark. For buildings, while there is always an apartment containing two given chambers, there may not be an apartment containing three given chambers. Sim- ilarly for modular lattices, the sublattice generated by three chains may not be distributive. This is the basic reason why the product in buildings and modular lattices is not associative. We do not know of any algebraic tools relevant to the study of non-associative LRBs. Remark. All four examples that we discussed share some common geometric properties. The LRB F (non-associative in the last two examples) is a simplicial (or polyhedral) complex. (Later in the paper, we will call such an F, a simplicial LRB.) Furthermore F is pure and gallery connected. Geometrically the projection FC is the chamber closest to C having F as a face. Also F G = ∩G≤D,D∈CFD for F, G ∈F.

3. Shellings and restriction maps Let ∆ be a ﬁnite pure d-dimensional simplicial complex. The term pure means that all maximal simplices have the same dimension. We will call the maximal simplices chambers. Let F be the set of all faces of ∆ and let C be the set of chambers of ∆. Also let ∆≥F be the simplicial subcomplex of ∆ consisting of all faces of the faces that contain F . Similarly C≥F stands for the set of chambers of 6 SWAPNEEL MAHAJAN

∆≥F . Let the partial order ≤ denote the face relation; that is, F ≤ G if F is a face of G. We say that ∆ is shellable if there is a linear order ≤S on the set of chambers C of ∆ such that for every D ∈ C except the ﬁrst in the linear order, we have

D ∩ (∪E

~~R(D)DD R(D)~~

~~Figure 1. The restriction map at work.~~

~~To a shelling ≤S, we can associate a restriction map R : C →F with R(D) deﬁned to be the face of D spanned by those vertices v of D for which D \ v ≤~~

~~D ∩ (∪E~~

(4) F = ⊔D∈C{F ∈F| R(D) ≤ F ≤ D}. Compare this statement with the restriction axiom (R2) in Section 4.1. Also note that R(D) = ∅, the empty face of ∆, if and only if D is ﬁrst in the linear order ≤S. To summarize, the restriction map R gives us the local data at every chamber associated with the shelling ≤S. It does not have enough information however to allow us to reconstruct ≤S. This is because the essence of a shelling is really a partial order. Here is what we can do. We can deﬁne a partial order ≤C on C as the transitive closure of the relation:

(5) E ≤C D if R(E) ≤ D. Observe that the chamber C by which we indexed our partial order is the one that occurred first in ≤S. The motivation for this notation will be more clear when we study the shelling axioms in Section 4.1. One can show directly that any linear extension of ≤C is a shelling of ∆ (compare this statement with the shelling axiom (S2)) and the shelling ≤S that we started with is one of them. We can also define r a more refined partial order ≤C on C as the transitive closure of the relation: r (6) E ≤C D if R(E) ≤ D and E is adjacent to D. r We will show (Lemma 2) that for a thin complex, the partial orders ≤C and ≤C on C are identical. The term thin means that every facet is contained in exactly 2 chambers. Now suppose instead that we start not with a shelling order ≤S but only with a map R : C →F that satisfies equation (4). Then can we say that R is the restriction map of a shelling? We know only a partial answer. The first thing to PROJECTIONS, SHELLINGS AND DUALITY 7 do is to define a relation ≤C using equation (5). (The chamber C is the unique chamber satisfying R(C) = ∅.) If the relation ≤C happens to be a partial order then it follows that any linear extension of ≤C is a shelling of ∆ and the restriction map of any of these is R. The above statements will be justified in the course of proving Theorem 1. For future use, we record three useful results about shellable complexes.

Lemma 1. Let F be any face of ∆ and let ≤S be a shelling of ∆. Then this linear order when restricted to C≥F is a shelling of ∆≥F . In other words, the link lk(F, ∆) of any face F in a shellable complex ∆ is again shellable with the induced order. ′ Proof. Let ≤S be a shelling of ∆ and let ≤S be its restriction to C≥F . By the deﬁnition of a shelling we have

~~(*) D ∩ (∪E~~

~~Claim: For D ≥ F , we have D ∩ (∪ ′ E)= ∪ ′ G. E~~

Lemma 2. Let ∆ be a thin shellable complex with shelling ≤S and restriction map r R. Let ≤C and ≤C be the partial orders on C deﬁned by equations (5) and (6) r respectively. Then the partial orders ≤C and ≤C on C are identical. r Proof. We only need to show that E ≤C D ⇒ E ≤C D, or equivalently, R(E) ≤ D r implies E ≤C D. We do this by constructing a gallery from E to D such that for consecutive chambers E′ and E′′ in the gallery, we have R(E′) ≤ E′′.

~~F E′′~~

~~E E′~~

~~Figure 2. A gallery from E to D in ∆≥F where F = R(E).~~

′ By Lemma 1, the shelling ≤S restricts to a shelling ≤S of ∆≥R(E). Hence ∆≥R(E) ′ is gallery connected at every stage of the shelling ≤S. And E is the starting chamber of the shelling. This allows us to choose a gallery from E to D in ∆≥R(E) ′ consistent with the shelling ≤S. This automatically implies consistency with the ′ ′′ original shelling ≤S. Let E and E be two consecutive chambers in the gallery 8 SWAPNEEL MAHAJAN

~~′ ′′ ′ ′′ such that E~~

Proposition 1. [9, Theorem 1.3] A ﬁnite d dimensional shellable complex ∆ is homotopy equivalent to a wedge of d-spheres. The number of these spheres is given by |{D | R(D) = D}|, where R is the restriction map associated with any shelling of ∆.

~~Note that if |{D | R(D) = D}| = 1 then the unique chamber D for which R(D)= D is the one that gets shelled in the end.~~

4. Axioms In this section we provide an axiomatic setup that relates projection maps, shellings and restriction maps. Motivation for some of the axioms was given in the previous two sections. The axioms are somewhat abstract and you may want to simply glance at them now. They are split into three categories. The first two axioms in each of the three categories really deal with a fixed chamber C ∈C. As we vary C ∈ C, it is natural to impose some compatibility condition. This is the content of the third (compatibility) axiom. It is easiest to swallow in the shelling case. We do not know of any way to make it more palatable in the other two cases. The main result of this section (Theorem 1) says that for a pure simplicial complex, the different sets of axioms are equivalent. The way to pass from one set of axioms to another is explained immediately after the statements of all the axioms. The examples mentioned in Section 2 are discussed from this axiomatic viewpoint in Section 6. It is a good idea to read that section before reading the proof of Theorem 1. Apart from serving as motivation, it will enable you to understand the geometric content of every step in the proof. Let ∆ be a finite pure simplicial complex. Let F be the set of all faces of ∆ and let C be the set of chambers of ∆. We always consider the empty face ∅ to be a face of ∆. Let the partial order ≤ denote the face relation; that is, F ≤ G if F is a face of G. Also let ⋖ stand for the cover relation; that is, G ⋖ D if G is a codimension 1 face of D. If D is a chamber and G ⋖ D then G is a facet of ∆.

4.1. The axioms. Before we state the projection axioms, we need a deﬁnition. Given a map F×C→C, we say that E is weakly C adjacent to D if they have a common face F such that FC = D. We may write this as D F E. Note that this is not a symmetric relation. Also we do not require F to be a facet. In fact, F could also be the empty face. This explains the term “weakly adjacent”. A weak C gallery is a sequence of chambers C1,...,Cn such that Ci+1 is weakly C adjacent to Ci for 1 ≤ i ≤ n − 1. We write this as C1 − ... − Cn. F1 F2 Fn−1 If more precision is required then we write C1 C2 ... Cn = D, where F1,...,Fn−1 is a sequence of faces such that, FiC = Ci and Fi ≤ Ci,Ci+1. It will be seen from axiom (P 1)(ii) that ∅C = C. Hence C can always be tagged on as the ﬁrst element of any weak C gallery. PROJECTIONS, SHELLINGS AND DUALITY 9

~~F 3~~

~~C 4 = D C 3 = F 3 C~~

~~C = F C C C = F C 2 2 F F 1 1 2 F 1~~

~~Figure 3. A weak C gallery C1 − C2 − C3 − C4.~~

Projection axioms (P ). For every F ∈F, there is a projection map C→C which we write C 7→ FC that satisfies (P1) (i) If FC = D then F ≤ D. (ii) If F ≤ C then FC = C. (iii) If FC = D and F ≤ G ≤ D then GC = D. (P2) If F ≤ D and GC = D for all F ≤ G ⋖ D then FC = D. ′ (P2 ) If F1C = F2C = D then there exists a face F ≤ F1,F2 such that FC = D. (P3) If FC = D and C1 − C2 − ... − Cn = D is any weak C gallery from C1 to D then FC1 = D. Axiom (P 1)(ii) implies that F = ∅ acts as the identity on C. Also note that it is a special case of axiom (P 1)(iii) obtained by setting F = ∅ and C = D. For axiom (P 2), as a special case when F = ∅, we get: If GC = D for all G ⋖ D then C = D. As another special case, when F = D ∈ C, the second condition holds vacuously and we get DC = D. In Figure 3, we have illustrated a weak C gallery. If the face F is such that FC = D then axiom (P 3) says that FC1 = D. The axiom (P 2′) is motivated by the theory of descent sets (see the discussion at the end of Section 6.2). In the course of proving Theorem 1, we will never use axiom (P 2′). The justification is the following simple lemma, whose proof we omit. Lemma 3. Let ∆ be a simplicial complex with a map F×C→C. Then ∆ satisfies (P 1) and (P 2) ⇔ ∆ satisfies (P 1) and (P 2′).

~~Restriction axioms (R). For every C ∈C, there is a restriction map RC : C→F that satisﬁes~~

~~(R1) For any C,D ∈C, we have RC (C)= ∅ and RC (D) ≤ D. (R2) For any C ∈C, we have F = ⊔D∈C{F ∈F| RC (D) ≤ F ≤ D}.~~

(R3) If RC (C1) ≤ C2,RC (C2) ≤ C3,...,RC (Cn−1) ≤ Cn = D then RC1 (D) ≤ RC (D) ≤ D. The symbol ⊔ in axiom (R2) stands for disjoint union. Also for simplicity of notation and suggestiveness, we will write the set of inequalities in the “if part” of RC (C1) RC (C2) RC (Cn−1) axiom (R3) as C1 C2 ... Cn. We will refer to this diagram as a sequence of C inequalities. In proving the equivalence of the axioms, we will see that a sequence of C inequalities is indeed a weak C gallery (and extremal in a certain sense). The ambiguity in notation will then disappear.

~~Shelling axioms (S). For every C ∈ C, there is a partial order ≤C on C that satisﬁes~~

~~(S1) For any F ∈F, the partial order ≤C restricted to C≥F has a unique minimal element. For the empty face F = ∅, this unique minimal element is C itself. 10 SWAPNEEL MAHAJAN~~

(S2) Every linear extension of ≤C is a shelling of ∆. ′ (S2 ) There exists a linear extension of ≤C that is a shelling of ∆. (S3) The partial orders are compatible in the sense that if D ≤C D1 ≤C D2 then D1 ≤D D2. Axiom (S2) clearly implies axiom (S2′). In the course of proving Theorem 1, we will see that we can drop axiom (S2) altogether and replace it by the weaker axiom (S2′). Hence the two axioms can be used interchangeably.

4.2. Connection between the axioms. We ﬁrst explain the basic idea. Projec- tion maps on ∆ give rise to many shellings of ∆. In fact we get a partial order (shelling) ≤C for every C ∈C as follows. We say E ≤C D if there is a common face F of the chambers D and E such that FC = E. This is illustrated in Figure 4. To be technically correct, the partial order ≤C is the transitive closure of the above relation.

~~D FC=E C~~

~~Figure 4. E occurs before D in the shelling associated to C.~~

To put it in words, among all chambers that contain F , the chamber that is smallest in the partial order ≤C is the chamber FC. And this is true for every F ∈F. The restriction map RC associated to ≤C can be deﬁned as follows. The face RC (D) is the smallest face F of D such that FC = D. We now record the above idea a little more formally. It is this formal connection between the axioms that we will use to prove Theorem 1. Going from

~~(P ) to (R): For every C ∈ C, we deﬁne a map RC : C→F. For D ∈ C, we let RC (D) be the face spanned by those vertices v of D which satisfy (D \ v)C 6= D.~~

(R) to (S): For every C ∈ C, we deﬁne a partial order ≤C to be the transitive closure of the relation: E ≤C D if RC (E) ≤ D. (Compare with equation (5).) (S) to (P ): For F ∈F and C ∈C, deﬁne FC to be the unique minimal element in C≥F with respect to the partial order ≤C . Here we used axiom (S1).

4.3. A shellable complex with a transitive group action. In Section 2 we discussed some examples from the viewpoint of projection maps. Now we consider an example that is more natural from the point of view of shellability. It remains to be seen whether this approach can be formalized to get more examples. Let ∆ be a simplicial complex of rank 2 that triangulates a circle; see Figure 5. For every C ∈ C, we deﬁne the partial order ≤C to be the total order on C that shells ∆ in the clockwise direction starting at C. The shelling axioms are easily checked. Note that there is a transitive action of Z/nZ on ∆, where n is the number PROJECTIONS, SHELLINGS AND DUALITY 11

~~FC F~~

~~R (D) C~~

~~E C~~

Figure 5. A non-metrical example. of edges in ∆. And this action is in some sense compatible with the partial orders ≤C . Using the connections between the three sets of axioms sketched above, we now describe the restriction and projection maps. Note that RC (C)= ∅ and RC (E)= E, where E is the chamber adjacent to C in the anticlockwise direction. For any other chamber D, RC (D) is the vertex of D that is further from C in the clockwise direction. For the projection maps, the only non-trivial case is when F is a vertex. If F ≤ C then FC = C. If not then we define FC to be the projection of C on F in the clockwise direction. Note that FC is not necessarily the closest chamber to C that contains F . In this sense ∆ is a non-metrical example. It is a good exercise to directly check the projection and restriction axioms. Remark. In the examples that we gave in Section 2, we always had a map F ×F → F. In our axiomatic setting, we may define such a map by using the projection maps C→C. For example, we can set F G = ∩G≤D,D∈CFD for F, G ∈F. However it is not clear what this would imply. For instance, we may ask whether we always get a LRB (non-associative included). In the example above, if we try to extend the projection maps to a product F×F→F, then ∆ is at best a non-associative LRB. 4.4. Main result. We now prove the main result of this paper. Theorem 1. Let ∆ be a finite pure simplicial complex. Then ∆ satisfies (P ) ⇔ ∆ satisfies (R) ⇔ ∆ satisfies (S). Proof. We show (P ) ⇒ (R) ⇒ (S) ⇒ (P ) using the connections between the axioms that we have already outlined. The proof is fairly routine. There is no particular reason why we choose this circle of implications. For instance, as an exercise, you may try to show (P ) ⇒ (S) directly. (P ) ⇒ (R). We verify axioms (R1), (R2) and (R3).

~~(R1). This is immediate from the deﬁnition of RC and axiom (P 1)(ii).~~

(R2). Let G be any facet of D. Then by the deﬁnition of RC , we have GC = D ⇔ RC (D) ≤ G. Now by axiom (P 2), we get RC (D)C = D. In fact RC (D) is the unique smallest face of D with this property. This follows from axiom (P 1)(iii) and the above “if and only of” statement. Hence {F ∈F | RC (D) ≤ F ≤ D} = {F ∈F| FC = D}. Axiom (R2) is immediate from this description.

RC (C1) RC (C2) RC (Cn−1) (R3). A sequence of C inequalities C1 C2 ... Cn = D is in fact a weak C gallery. This is due to the fact that RC (Ci)C = Ci. Also RC (D) is 12 SWAPNEEL MAHAJAN such that RC (D)C = D. Hence applying axiom (P 3) to the above weak C gallery with F = RC (D), we get RC (D)C1 = D. Since RC1 (D) is the smallest face F such that FC1 = D, we get RC1 (D) ≤ RC (D) ≤ D.

(R) ⇒ (S). We ﬁrst show that ≤C is a partial order. For this we will use all three restriction axioms. The only thing to check is antisymmetry. Expanding out the deﬁnition of ≤C that we have given, C1 ≤C Cn if we have a sequence of RC (C1) RC (C2) RC (Cn−1) C inequalities C1 C2 ... Cn. To check antisymmetry, we show that a (non-trivial) sequence as above cannot close in on itself. Suppose not, then we have a minimal circular sequence,

~~RC (C1) RC (C2) RC (Cn−1) RC (Cn) C1 C2 ... Cn Cn+1 = C1, which is shown schematically in Figure 6. Thenby axiom (R3), we obtain RC1 (Cn) ≤~~

~~RC (Cn) ≤ Cn. However by axiom (R1) we also have ∅ = RC1 (C1) ≤ RC (Cn) ≤ C1. Since C1 6= Cn, this contradicts axiom (R2). This shows that ≤C is a partial order.~~

~~R (C ) C2 C3 C 1~~

~~R (C ) C n~~

~~Figure 6. A situation that can never occur.~~

Remark. In terms of projection maps, RC (Cn)C1 is forced to be both C1 and Cn, which shows why the situation in Figure 6 never occurs. Now we verify axioms (S1), (S2) and (S3). (S1). By axiom (R2), for any F ∈ F, there is a unique chamber D such that RC (D) ≤ F ≤ D. For any other chamber E containing F , we have RC (D) ≤ E. Hence by deﬁnition of ≤C , we have D ≤C E. This shows that D is the unique minimal element for ≤C when restricted to C≥F .

(S2). Let ≤S be any linear extension of ≤C . Since RC (C)= ∅, we know that C is the unique smallest element in the partial order ≤C . To show that ≤S is a shelling, it is enough to prove the following.

~~Claim. For any D ∈C such that D 6= C, we have D ∩ (∪E~~

RC (D) (S3). If D ≤C D1 ≤C D2 then we have a sequence of C inequalities D ... − RC (D1) RD (D1) D1 ...−D2. Then axiom (R3) implies that D1 ...−D2 is a sequence of D inequalities. Note that we have replaced C by D. So D1 ≤D D2. PROJECTIONS, SHELLINGS AND DUALITY 13

(S) ⇒ (P ). Now we verify axioms (P 1), (P 2) and (P 3). For this we will use the weaker axiom (S2′) instead of (S2). (P 1). (i) This follows directly from the definition of the product. (iii) Here we assume that FC = D and F ≤ G ≤ D. The first assumption says that D is the unique minimal element in the partial order ≤C restricted to C≥F . Since F ≤ G ≤ D, it follows that D is also the unique minimal element in the partial order ≤C restricted to C≥G. Hence GC = D. As mentioned before, axiom (P 1)(ii) is a special case of (P 1)(iii). ′ (P 2). Using axiom (S2 ), we choose a linear extension ≤S of ≤C which is a shelling of ∆. By Lemma 1, this linear order when restricted to C≥F gives a shelling of ∆≥F , and similarly of ∆≥G for all F ≤ G ⋖ D. Our assumption GC = D for all F ≤ G ⋖ D says that for all such G, the shelling order restricted to C≥G has D as the first element. If we assume that FC 6= D then D is not the first element in the shelling order restricted to C≥F . Hence D ∩ (∪E

(P 3). If D and E are weakly C adjacent then D ≤C E. Hence if C1−C2−...−Cn = D is a weak C gallery then C1 ≤C D. Let F be such that FC = D. Now if E ∈C≥F then D ≤C E. Therefore C1 ≤C D ≤C E. Applying axiom (S3), we get D ≤C1 E for all E ∈C≥F . Hence FC1 = D.

~~From now on, if ∆ is a pure simplicial complex that satisﬁes (P ), or equivalently, (R) or (S), then we will simply say that ∆ satisﬁes our axioms.~~

5. An almost example The purpose of this section is to identify a potential source of examples that satisfy our axioms. In Section 4 the poset F was the face lattice of a simplicial complex ∆. It is natural to consider a more general situation, where instead of ∆, we just have a poset F with the maximal elements playing the role of C. The projection axioms still make sense though the role they now play is not exactly clear. An interesting case to consider is when F is a LRB. We will show that a LRB satisﬁes axioms (P 1) and (P 3). However it can easily violate axiom (P 2). We ﬁrst discuss this situation. We will see that axiom (P 2′) is better suited for posets than axiom (P 2). See also the remark at the end of Section 6.2.

~~123 132 213 231 312 321 CC1 2~~

~~12 13 21 23 31 32~~

~~SS1 2 1 2 3~~

~~F φ~~

~~Figure 7. LRBs that violate axiom (P 2). 14 SWAPNEEL MAHAJAN~~

5.1. On why a LRB violates axiom (P 2). To see what goes wrong, consider the following example. Let S1 and S2 be any two LRBs. Consider the LRB S shown in Figure 7 (on the left) with F1F2 = C1 and F2F1 = C2 for F1 ∈ S1 and F2 ∈ S2. We ﬁrst check that S is indeed a LRB. Then by deﬁnition GC1 = C2 for F < G ⋖ C2. However FC1 = C1 6= C2. Hence S violates axiom (P 2). We also see that S violates axiom (P 2′). To give a concrete example that illustrates the same problem, consider the free LRB with identity on n generators, denoted Fn. The elements of Fn are sequences x = (x1,...,xl) of distinct elements of the set [n] = {1,...,n}, 0 ≤ l ≤ n. We multiply two such sequences by

~~(x1,...,xl)(y1,...,ym)=(x1,...,xl,y1,...,ym) , where the hat means “delete any element that has occurred earlier”.b For example, (21)(35416) = (213546).~~

Figure 7 (on the right) shows the Hasse diagram of F3, the free LRB on 3 generators. Check that Fn violates axiom (P 2). Also note that every chamber has only one ′ facet. However note that Fn does satisfy axiom (P 2 ). These examples show that a LRB is not a geometric object in any sense of the term. A natural question that comes up is whether a simplicial LRB always satisfies axiom (P 2). We do not know the answer. For the LRBs that we considered in Section 2, the answer is yes. In other words, the LRB associated to any hyperplane arrangement satisfies axiom (P 2) (and hence the same holds for flag (order) complex ∆(M) of a distributive lattice M). It is an easy and illuminating exercise to check this directly using the definition of the product involving sign sequences. 5.2. A LRB satisfies axioms (P 1) and (P 3). Axiom (P 1) follows directly from the definition of a LRB and axiom (P 3) holds by the following proposition. It is a good exercise to check axiom (P 3) directly for the free LRB. Proposition 2. Let F be a poset with an associative product F×F→F. Let C, the set of maximal elements of F, be a left ideal in F. Also assume that F satisfies axiom (P 1) with the additional property that F ≤ FF ′ for any F,F ′ ∈F. Then F satisfies axiom (P 3). Proof. Though our setting is somewhat abstract, it would help to keep in mind the F1 F2 Fn−1 picture in Figure 3. Let C1 C2 ... Cn = D be a weak C gallery; that is, FiC = Ci and Fi ≤ Ci,Ci+1. We want to show that if FC = D then FC1 = D. For notational consistency, we put F = Fn. Let G = {H ∈F| Fn ≤ H ≤ Cn = FnC}. We need to show that FnC1 = Cn. We first note three easy consequences of parts (i), (ii) and (iii) respectively of axiom (P 1). (1). If Fn ≤ H and HE = Cn for some E ∈C then H ∈G. (2). For 2 ≤ i ≤ n, we have Fi−1FiC = FiC. This is because Fi−1 ≤ Ci = FiC. (3). If H ∈G then HC = Cn.

We now claim that FnFi ∈G for 1 ≤ i ≤ n. Proof of the claim. We do a reverse induction on the index i. Clearly FnFn ∈ G. To do one more step, note that FnFn−1 ∈G. This is because Fn ≤ FnFn−1 by our additional assumption and FnFn−1 ≤ Cn follows from Fn−1,Fn ≤ Cn and axiom (P 1). Now assume that FnFi ∈G for some i. We show that FnFi−1 ∈G. PROJECTIONS, SHELLINGS AND DUALITY 15

Using (3) we get (FnFi)C = Cn. Rewriting this as Fn(FiC) = Cn and using (2), we get Fn(Fi−1FiC) = Cn which we write as (FnFi−1)FiC = Cn. Note that Fn ≤ FnFi−1. Hence we apply (1) with H = FnFi−1 and E = FiC to conclude that FnFi−1 ∈G. This completes the induction step and the claim is proved. From the claim we get FnF1 ∈ G. Using (3) we have (FnF1)C = Cn which is same as Fn(F1C)= FnC1 = Cn. This is exactly what we wanted to show.

The discussion in this section shows that the LRBs in Section 2 satisfy the projection axioms and hence are examples of our theory. In the next section, we again consider these examples (including the non-associative ones) but from a more intuitive and geometric perspective.

6. The metric setup In this section we show that if a chamber complex ∆ satisfies the gate property then it satisfies our axioms. This is essentially an axiomatic restatement of a result of Scharlau; see the proposition in [27, Section 3]. (He states it in terms of shellings. Also he does not consider the compatibility axiom.) Furthermore in this special situation, the maps of Section 4 have a geometric meaning. To make this clear we first need some definitions.

6.1. Some deﬁnitions. Let ∆ be a pure simplicial complex. The maximal simplices are called chambers. We say two chambers are adjacent if they have a common codimension 1 face. A gallery is a sequence of chambers such that consecutive chambers are adjacent. We say that ∆ is gallery connected if for any two chambers C and D, there is a gallery from C to D. For any C,D ∈ C, we then deﬁne the gallery distance d(C,D) to be the minimal length of a gallery connecting C and D. And any gallery which achieves this minimum is called a geodesic gallery from C to D. Another relevant metric notion is the following.

~~Gate Property. For any face F ∈ F and chamber C ∈ C, there exists a chamber D ∈ C≥F such that d(C,D) ≤ d(C,E) for any E ∈ C≥F . Furthermore d(C,E) = d(C,D)+ d(D,E).~~

The gate property implies that ∆ is strongly connected; that is, C≥F is gallery connected for all F ∈ F. In fact it implies that C≥F is a convex subset of C; that is, if D and E are any two chambers in C≥F then any geodesic gallery from D to E lies entirely in C≥F .

~~D D2 C~~

~~Figure 8. A geodesic gallery that illustrates shelling compatibility. 16 SWAPNEEL MAHAJAN~~

6.2. Geometric descriptions of the relevant maps. We now use the above metric notions to define the maps that played a role in the axiomatic setup of Section 4. Let ∆ be a chamber complex. It is by definition a gallery connected pure simplicial complex. For every C ∈ C, we define a partial order ≤C . We say that D ≤C E if there is a geodesic gallery from C to E that passes through D. With this definition of the partial orders ≤C , we analyze the shelling axioms. We first note that axiom (S3) is a consequence of our definition. If D ≤C D1 ≤C D2 then we have a geodesic gallery C − ... − D − ... − D1 − ... − D2 as shown in Figure 8. This restricts to a geodesic gallery D − ... − D1 − ... − D2, which implies D1 ≤D D2. Now we look at axiom (S1). It says that for any C ∈C and E ∈C≥F there is a unique chamber D ≥ F such that D ≤C E; that is, such that there is a geodesic gallery C − ... − D − ... − E. This is equivalent to saying that ∆ has the gate property. Next we claim that a chamber complex ∆ with the partial orders ≤C as defined above satisfies the shelling axioms (S) if and only if it has the gate property. From what we have so far, we only need to show that the gate property implies axiom (S2). This can be checked directly. However we will prove this using the projection axioms.

Remark. If ∆ = Σ is the Coxeter complex associated to a Coxeter group W then ≤C coincides with the weak Bruhat order on W , after we choose C as the fundamental chamber and identify chambers of Σ with elements of W . Now we assume that ∆ has the gate property. We show that this implies the projection axioms. Using the gate property, we define FC to be the chamber containing F that is closest to C. Axiom (P 1) follows. The gate property further says that a weak C gallery C1 − C2 − ... − Cn can be extended to a geodesic gallery C − ... − C1 − ... − C2 − ... − Cn. Axiom (P 3) now follows. To show axiom (P 2), we proceed by contradiction. Let FC = E 6= D. Choose a geodesic gallery C − ... − E − ... − D. Let G be the facet of D that is crossed by this gallery in the final step. Then GC 6= D, which is a contradiction. To summarize, ∆ satisfies the projection axioms (P ) (with the closest chamber definition) if and only if it satisfies the gate property. To complete the story we now describe the restriction maps. Recall from Sec- tion 4.2 that RC (D) is the face of the chamber D spanned by vertices v such that (D \ v)C 6= D. It can be equivalently described as the face of the chamber D spanned by vertices v which have the following property. There is a minimal gallery from C to D that passes through the facet D \ v in the final step. We refer to the second description as Des(C,D). The equivalence of the two descriptions says that RC (D) = Des(C,D). The terminology Des(C,D) is again motivated by the theory of Coxeter groups. Des(C,D) is the face of D spanned by “the descent set of D with respect to C”. For an explanation, see the appendix by Tits to Solomon’s paper [30] or the more elaborate exposition in [14, Section 9]. Remark. Another way to describe Des(C,D) is as the smallest face F of D such that FC = D. Note that this definition makes sense for any LRB that satisfies ′ axiom (P 2 ). And we can define the partial order ≤C using the notion of a weak C gallery. We have proved the following theorem. PROJECTIONS, SHELLINGS AND DUALITY 17

Theorem 2. Let ∆ be a chamber complex. Also let the partial orders ≤C , the restriction maps RC and the projection maps be as defined in this section. Then ∆ satisfies our axioms if and only if it satisfies the gate property. 6.3. Examples of Section 2 revisited. Hyperplane arrangements, buildings and modular lattices are all examples of chamber complexes that satisfy the gate property. Hence by Theorem 2 we have established that they indeed satisfy our axioms. For oriented matroids (in particular, central hyperplane arrangements), the gate property is due to Björner and Ziegler and for buildings it is due to Tits [33]. The gate property for modular lattices (in particular, distributive lattices) was proved by Abels, see [1, Proposition 2.9]. Alternatively, for distributive lattices, we can deduce the gate property by combining the following facts. If ∆ satisfies the gate property then so does any subcomplex ∆c spanned by a convex subset of chambers. The flag (order) complex ∆(M) of a distributive lattice M corresponds to a convex set of chambers in ∆(Bn), the complex of the braid arrangement. More on Example 1. The metric notions that we described in this section can be made very explicit for this example. We prefer to restrict to the simplicial case though all statements except the last make sense and hold in general. A minimal gallery from C to D crosses exactly those hyperplanes that separate C and D. Consequently the gallery distance d(C,D) is given by the number of hyperplanes separating C and D. We have D ≤C E if and only if the hyperplanes that separate C and D also separate C and E. The chamber FC is the unique chamber containing F for which no hyperplane passing through F separates C from FC. Also RC (D) is the face of D whose support is the intersection of those walls of D that do not separate D from C. Rigorous proofs of these unjustified facts can be found in any of the references cited earlier or you may accept them as “intuitively obvious”.

7. Type selected subcomplexes In this section, we introduce the notion of a local flag h vector. This notion will also be used to motivate the discussion in the next section. For general background on the flag h vector, see [34, Section 8.3]. We will present some well-known results as well as some new ones. However our approach will be different and will use projection maps. The treatment will be fairly self-contained. A labeling of a pure simplicial complex ∆ by a set I is a function which assigns to each vertex an element of I, in such a way that the vertices of every chamber are mapped bijectively onto I. A labeled simplicial complex is also sometimes called “numbered” or “balanced”. Let ∆ be a labeled simplicial complex and let ∆J be its type selected subcomplex consisting of faces whose label set or type is contained in J. In Section 7.1, we study the connection between the homotopy type of ∆J and the flag h vector. In Section 7.2, we define the local flag h vector. This is the dual picture for the flag h vector. The fact that this viewpoint is useful will be seen from the example that we present later in Section 7.4. We will also consider the unlabeled case briefly in Section 7.3. We start with the following result of Björner that generalizes Proposition 1. Proposition 3. [9, Theorem 1.6] For a labeled shellable complex ∆, the type selected subcomplex ∆J is shellable and homotopy equivalent to a wedge of (|J|−1)-spheres. 18 SWAPNEEL MAHAJAN

The number of these spheres is given by βJ = |{D | R(D) has type J}|, where R is the restriction map associated with the shelling. Remark. The proposition is proved by showing that the shelling order for ∆ “in- duces” a shelling order for ∆J . This raises the following question. Suppose we assume that ∆ satisfies our axioms with RC as the restriction maps. Also let RJ,C be the maps for ∆J induced by RC . Then what kind of restriction axioms do the induced maps RJ,C satisfy for ∆J ? The axioms need to be suitably generalized because there is more than one restriction map for every chamber in ∆J . 7.1. The restriction map and the flag h-vector. We fix a chamber C ∈ C and assume that ∆ satisfies the first two axioms (in each category) for C. Recall that RC (D) is the unique smallest face F of D such that FC = D. Proposition 3 applies to our situation because of axiom (S2) which implies that ∆ is shellable. The restriction map R in the proposition is the map RC , where C is the chamber that we have fixed. Next define

~~βJ = |{D | RC (D) has type J}|.~~

We will show that the vector (β(J))J⊆I coincides with the “flag h-vector” defined below. This result goes back to Björner or even earlier. First we define the flag f-vector of ∆ by setting fJ (∆) equal to the number of simplices of type J. The flag h-vector is then obtained by writing

(7) fJ (∆) = hK (∆), KX⊆J or, equivalently, |J−K| (8) hJ (∆) = (−1) fK (∆). KX⊆J Proposition 4. Let ∆ be as above and let I be the set of types of vertices. Then for any J ⊆ I, βJ = hJ (∆).

Proof. Let FJ be the set of simplices of type J. There is a 1–1 map FJ →C, given by F 7→ FC, where C is fixed. It is 1–1 because we can recover F from FC as the face of type J. Its image is the set of chambers D for which the type of RC (D) is contained in J. Hence fJ (∆) = βK . KX⊆J The proposition now follows from (7). Remark. The above material is taken from [14]. The difference is that Ken Brown stated this proposition for Coxeter complexes. Also he worked with descent sets rather than restriction maps. We know from the discussion in Section 6 that in the metric setup, RC (D) = Des(C,D). 7.2. The dual picture. Now we assume that ∆ satisfies all three axioms. In other words, we are now allowed to vary C ∈C and have the compatibility axiom. Note that βJ does not depend on the choice of the chamber C. For example, from Proposition 3, we know that βJ counts the number of spheres in ∆ and hence is independent of the choice of C. Also from Proposition 4, we know that it coincides PROJECTIONS, SHELLINGS AND DUALITY 19 with the flag h vector of ∆. We also see that hJ (∆) = |{D | RC (D) has type J}|. This gives geometric interpretations of the flag h vector, one for every C ∈C. Now we make full use of the compatibility axiom by turning the picture around. For the map F×C→C that maps (F,C) to FC = D, we fix D and vary C. More precisely, we define the local flag h vector hJ (D) = |{C | RC (D) has type J}|. Similarly define the local flag f vector fJ (D) to be the number of chambers C such that FC = D, where F is the face of D of type J. This can be rewritten as fJ (D)= |{C | The type of RC (D) is contained in J}|. Now observe that

~~(9) fJ (D)= hK (D). KX⊆J~~

The numbers fJ (D) and hJ (D) do in general depend on D. If we average them over all D ∈ C then we recover the usual flag vectors. To see this, let σJ be the sum of all faces of type J. In particular, σI is the sum of all chambers. Note that σJ σI = D∈C fJ (D)D. By counting the total number of terms involved, 1 we get fJ (∆) =P |C| D∈C fJ (D). This along with (7) and (9) says that hJ (∆) = 1 P |C| D∈C hJ (D). Note that it is not obvious from the definition that the right hand sideP is an integer. The numbers hJ (D) give us a different way to understand the flag h vector. A natural question that arises is: When is hJ (∆) = hJ (D) for all D ∈ C? We will address this in more detail in Section 8.

7.3. The unlabeled case. If ∆ is not labeled then we work with the ordinary f and h vectors rather than the flag vectors. Let ∆ be a simplicial complex of rank n that satisfies our axioms. As before, we will need the compatibility axiom only when we pass to the dual picture. We define βj = |{D | RC (D) has rank j}|. The unlabeled version of Proposition 3 is as follows.

Lemma 4. For a shellable complex ∆ of rank n, the (k − 1)-skeleton ∆k−1 is shellable and homotopy equivalent to a wedge of (k − 1)-spheres. The number of k n−i−1 these spheres is given by i=0 k−i βi. P The fact that shellability is inherited by k-skeleta appears as Corollary 10.12 in [11]. Also see the references cited therein. The formula for the number of spheres was told to me by Björner. As before, we can show that the vector (βj )0≤j≤n coincides with the “h-vector” defined below. First we define the f-vector of ∆ by setting fj(∆) equal to the number of simplices of rank j. The h-vector is then obtained by writing j n − k (10) fj(∆) = hk(∆). n − j kX=0 Proposition 5. Let ∆ be as above. Then for any 0 ≤ j ≤ n,

~~βj = hj(∆).~~

Proof. Let Fj be the set of simplices of rank j. As before, we consider the map Fj → C, given by F 7→ FC, where C is fixed. Recall that for a fixed C and D, {F ∈F | RC (D) ≤ F ≤ D} = {F ∈F | FC = D}. This says that the image of the above map is the set of chambers D for which the rank of RC (D) is less than or equal to j. However this map is no longer 1–1. The number of times a fixed 20 SWAPNEEL MAHAJAN

n−k chamber D occurs in the image is n−j . This is the number of faces of D of rank j that contain RC (D) (whose rank we set equal to k). This gives j n − k fj (∆) = βk n − j kX=0 and the proposition now follows.

To get the dual picture, let hj (D) = |{C | RC (D) has rank j}|. Similarly let fj(D) = |{(F,C) | FC = D, rk(F ) = j}|. As in the proof of Proposition 5, j n−k fj(D)= k=0 n−j hk(D). Now if we let σj be the sum of all faces of rank j then 1 σjσn = PD∈C fj(D)D. This gives fj(∆) = |C| D∈C fj(D) and then hj (∆) = 1 P P |C| D∈C hj(D). P Remark. For a labeled complex, we have fj = |J|=j fJ ,hj = |J|=j hJ and σj = |J|=j σJ . So in this case, the results for thePf and h vector followP from the correspondingP result for the flag vectors. For instance, using the first two equations, one can recheck that equation (7) reduces to equation (10). 7.4. Simplicial hyperplane arrangements. Let ∆ be a simplicial hyperplane arrangement. We now give a description of the local flag h vector hJ (D) in this case. This dual description given in Section 7.2 is easier to visualize than the usual one in Section 7.1. Recall that in this situation, RC (D) is the face of D whose support is the intersection of those walls of D that do not separate D from C; see the description at the end of Section 6. Using this description, in the labeled case, we have the following lemma.

Lemma 5. The local ﬂag h vector hJ (D) is the cardinality of the set of chambers C such that D and C lie on the same side of a wall of D if and only if the wall contains the face of type J of D. The local ﬂag f vector fJ (D) is the cardinality of the set of chambers C such that D and C lie on the same side of all walls of D that contain the face of type J of D.

~~t t s s D D hs (D) u h (D) fs,t(D) s,t~~

~~Figure 9. The schematic picture in rank 3.~~

In Figure 9 only the three walls of the chamber D have been drawn. The only chamber that is not seen in the picture is the one directly opposite to D. The local flag f vector for D is represented by various lunes formed at D by its faces. In the figure on the left, the shaded part shows the lune for the face of D of type {s,t} and corresponds to fs,t(D). The local flag h vector for D is represented by the 8 regions into which the three walls divide the sphere. We have marked the region that corresponds to hs(D) and shaded the one for hs,t(D) in the figure on the right. PROJECTIONS, SHELLINGS AND DUALITY 21

In the unlabeled case, hj (D) is the cardinality of the set of chambers C such that D and C lie on opposite sides of exactly j of the n walls of D. In Figure 9, for example, h1(D) counts the chambers that lie in the three regions adjacent to D along its facets. Remark. If ∆ is a finite Coxeter complex or a finite (spherical) building associated with a BN-pair then the local flag vectors coincide with the usual or global flag vector. Also see the remark after Proposition 6.

Next we give a tiny application of this description. Let ∆min be the Coxeter complex of the reﬂection arrangement xi = 0, where 1 ≤ i ≤ n. In fact Figure 9 n as drawn corresponds to the case n = 3. Observe that hj(D) = j . From our interpretation, it is clear that for any other simplicial arrangement ∆ of rank n, we n n get hj(D) ≥ j for all D ∈C and hence hj(∆) ≥ j . ′ ′ Now consider the barycentric subdivision ∆ of ∆. Then ∆min corresponds to the Coxeter arrangement of type Bn, namely, xi = ±xj and xi = 0, where 1 ≤ i< j ≤ n. However in general, ∆′ is not a hyperplane arrangement. It is just a labeled ′ ′ simplicial complex. Hence we cannot directly conclude that hJ (∆ ) ≥ hJ (∆min) (which is known to be true). This raises an important question. If ∆ satisﬁes our axioms then what can we say about its barycentric subdivision ∆′ ? For a positive result in this direction, see [8, Theorem 5.1] which says that if ∆ is shellable then so is ∆′.

~~8. Commutativity issues~~

In this section, we address the question raised in Section 7: When is hJ (D) = hJ (∆) for all D ∈ C? This can be rephrased as a commutativity problem. Recall from the previous section that σJ (resp. σj) is the sum of all faces of type J (resp. rank j). Also hJ (D) and hJ (∆) are the local and global versions of the flag h vector of ∆. And hj(D) and hj (∆) are the corresponding versions of the h vector of ∆. Proposition 6. Let ∆ be a simplicial LRB (possibly non-associative) that satisfies our axioms. Then (labeled case) hJ (∆) = hJ (D) for all D ∈ C and for all subsets J ⊆ I ⇔ σI σJ = σJ σI for all subsets J ⊆ I. (unlabeled case) hj(∆) = hj (D) for all D ∈C and for all 0 ≤ j ≤ n ⇔ σnσj = σjσn for all 0 ≤ j ≤ n. Proof. We do only the labeled case. The unlabeled case is similar. Suppose that σI σJ = σJ σI for all subsets J ⊆ I. Counting the number of times a fixed chamber D occurs on both sides of the equality, we get fJ (∆) = fJ (D), or equivalently, hJ (∆) = hJ (D). Remark. In the proposition above, a concrete case to keep in mind is that of simplicial hyperplane arrangements. Also we required ∆ to satisfy our axioms because we only defined hJ (D) in that situation. And we required ∆ to be a LRB so as to make sense of the product σI σJ . Of course, we only needed to use that CF = C for C ∈C and F ∈F. If ∆ is a finite Coxeter complex or a finite (spherical) building associated with a BN-pair then there is a type and product preserving group action on ∆ and it is transitive on C. Hence the commutativity condition in Proposition 6 is automatic. And so for this case we obtain that hJ (D)= hJ (∆) for all D ∈C. This raises the 22 SWAPNEEL MAHAJAN following question: Let ∆ be a thin labeled chamber complex of rank greater than 3 such that σI σJ = σJ σI for all subsets J ⊆ I. Then is it true that ∆ is a Coxeter complex ? 8.1. Connection with random walks. Bidigare, Hanlon and Rockmore [5] found a natural family of random walks associated with hyperplane arrangements. These walks were studied further by Brown and Diaconis [15] and generalized to LRBs by Brown [14]. One reason this development is exciting is that the walks admit a rather complete theory. We now describe the walk. Using the projection operators, define the following walk on the chambers C of ∆: If the walk is in chamber C, choose a face F of rank j at random and move to the projection D = FC. (We assume here that the faces of rank j are chosen uniformly.) Note that this random walk on chambers has a uniform stationary distribution if and only if σnσj = σjσn. Here n is the rank of ∆. More generally, we can run a random walk on C using a probability distribution {wF }F ∈F on F, the faces of ∆. Also if the complex is labeled then we can define the walk by choosing instead a face F of type J at random. Then the walk on chambers has a uniform stationary distribution if and only if σI σJ = σJ σI . However for the rest of the section, we will restrict to the unlabeled case. 8.2. Some commutativity conditions. Motivated by the above discussion, we look at a more general commutativity problem. We work with a graded LRB. All known examples of LRBs are graded. However we do not know whether this follows from the definition. For now, we do not make any further assumptions. Let S be a LRB, L its associated lattice and supp: S ։ L the support map. For any X ∈ L, let S≤X = {y ∈ S : supp y ≤ X}. Then one can check that S≤X is also a LRB whose associated lattice is the interval [0ˆ,X] in L. Here 0ˆ is the support of the identity of S and hence the minimum element of L. Also the set of chambers C consists precisely of those elements of S whose support is 1,ˆ the maximum element of L. We say that a LRB satisfies the commutativity condition for i and j if

(Ci,j ) σiσj = σjσi. Similarly we say that it satisﬁes the uniformity condition (U) if for any i,j the number of times a chamber occurs in the product σiσj does not depend on the chamber we choose. Loosely speaking, these conditions may be thought of as certain symmetry conditions on the LRB. We have already given some motivation for the condition (Ci,j ). The motivation for condition (U) becomes clear from the following proposition.

Proposition 7. Let S be a LRB such that S≤X satisfies the uniformity condition (U) for all X ∈ L. Then S (and hence all S≤X ) satisfies the commutativity condition (Ci,j ) for all i,j. Proof. We first observe that supp(F G) = supp(GF ) = supp F ∨ supp G. In particular, for H = F G, we have supp(F ), supp(G) ≤ supp(H). Hence the coefficient of H in σiσj remains unchanged if we replace S by the subsemigroup S≤supp(H). This shows that if S satisfies (Ci,j ) then so does S≤X for all X ∈ L. Now we show that S (and hence all S≤X ) satisfies (Ci,j ) for all i,j. We first check that for any D ∈ C, the coefficient of D in σiσj is same as the coefficient of D in σjσi. Since we assume that S satisfies (U), we simply need to check that PROJECTIONS, SHELLINGS AND DUALITY 23

|{(F, G) | rk(F )= i, rk(G)= j,FG is a chamber}| = |{(G, F ) | rk(F )= i, rk(G)= j,GF is a chamber}|. This is true since F G is a chamber ⇔ GF is a chamber. This is because of our earlier observation that supp(F G) = supp(GF ). To do the general case, we need to check that for any H ∈ F, the coefficient of H in σiσj is same as the coefficient of H in σjσi. To do this computation, we replace S by S≤supp(H). Note that H is a chamber in S≤supp(H) and by our assumption S≤supp(H) satisfies (U). Hence we now apply the previous argument and the proposition is proved.

Corollary. If S is the Coxeter complex of type An−1 or Bn, then S satisﬁes (Ci,j ) for all i,j. Proof. The uniformity condition (U) is automatic for a Coxeter complex Σ because the Coxeter group acts transitively on the chambers of Σ. However to apply the above proposition, we require (U) to hold for all S≤X . For types An−1 or Bn, given any X ∈ L, S≤X is again isomorphic to a Coxeter complex of the same type but now with a smaller value of n. Hence the Coxeter complex of type An−1 or Bn satisﬁes (U) for all S≤X and we can apply the proposition.

Remark. For types An−1 and Bn, the semigroups and the underlying lattices can be made combinatorially explicit [14]. The fact about S≤X being a Coxeter complex for types An−1 or Bn that we used in the proof above is then quite elementary. For the case when X is a hyperplane (wall), this problem has been treated in general for any Coxeter complex by Abramenko [2]. A completely diﬀerent proof of this corollary can be given using the language of card shuﬄes, see [3, 20]. We also note that for S, the Coxeter complex of type Dn, the subcomplexes S≤X are not even Coxeter complexes. This is the primary reason why the complex of type Dn fails the commutativity condition in general. We have checked by explicit computation that there exists an i,j such that the Coxeter complex of type D4 does not satisfy (Ci,j ). 8.3. A low rank computation. In contrast to the previous subsection, we get a less restrictive result if we work with low ranks.

~~Theorem 3. Let S be an oriented matroid of rank 3. Then S satisfies C1,2 ⇔ S satisfies C1,3 ⇔ S satisfies C2,3 ⇔ S is simplicial.~~

Proof. A part of this theorem, namely, S satisfies C1,3 ⇔ S is simplicial occurs as Proposition 1 in [6]. It says that the random walk on C driven by a uniform probability distribution on the vertices of S has a uniform stationary distribution iff S is simplicial (see the discussion in Section 8.1). Here we will prove that S satisfies C1,2 ⇔ S is simplicial. The other cases are simpler and we leave them out. The spirit of the computation is as in [6]. Let v,e and f be the number of vertices, edges and faces respectively in S. We need to look at σ1σ2 = σ2σ1. Note that a vertex can never occur in either product. And the number of times an edge appears on either side is the same. (This is because we can restrict to the support of that edge, which is just a pseudoline.) So the non-trivial case is that of chambers. We fix a chamber D. Denote the chamber exactly opposite to D by D. We count the number of times that either D or D occurs in σ1σ2 and σ2σ1. (It will be evident from the computation as to why we group D and D together. Note that due to the antipodal symmetry of the arrangement, we do not lose anything essential by doing this.) 24 SWAPNEEL MAHAJAN

~~Figure 10. The walls of a 5-gon.~~

If D is a k-gon then consider the k walls of D (and hence D). We call an edge of S an interior edge if it does not lie on any of these k walls. Figure 10 illustrates the case when k = 5. One of the walls has been highlighted. The only chamber that is not seen in the picture is the pentagon D. It is on the backside bounded by the five segments that bound the figure. We first look at σ1σ2 and count the number of times that either D or D occurs in this product; that is, |{(F, G) | F G = D or F G = D, rk(F ) = 1, rk(G) = 2}|. We use the geometry of the arrangement as illustrated in the figure. Note that F must be a vertex of D or D. We consider two cases. In the first case, let G be an interior edge or an edge of either D or D. Then there are k − 2 choices for F . The individual contributions of D or D depend on the location of the edge G, however the net contribution is always k − 2. In the second case, G lies on one of the k walls but is not a face of either D or D. Now there are k − 3 choices for F . Hence the required count is (k − 2)e − (number of edges on all the walls) + 2k. Now we look at σ2σ1. We want to count |{(G, F ) | GF = D or GF = D, rk(F )= 1, rk(G) = 2}|. The only G’s that we need consider are the edges of D and D. Given a wall H of D (and D), the edge of D that lies on H and its opposite edge (which is an edge of D) together can pair with all vertices except the ones that lie on H. Summing up over all the k walls, we get the count to be kv − (the sum of the no. of vertices on each of the k walls). Now we compare the counts for σ1σ2 and σ2σ1. Since the no. of vertices is same as the no. of edges on each wall (since it is a pseudoline), we get σ1σ2 = σ2σ1 ⇔ (k − 2)e + 2k = kv ⇔ kf = 2e. The last equality holds if and only if the arrangement is made entirely of k-gons. However, by Levi’s theorem [18, p. 25]), any arrangement must have at least one triangle. Hence we get k = 3.

9. The opposite or duality axiom Let ∆ be a simplicial complex of rank n. Under suitable hypotheses on ∆ (for example, if ∆ triangulates a sphere), the h vector of ∆ satisfies the linear relations hi = hn−i for 0 ≤ i ≤ n. Similarly if ∆ is a labeled simplicial complex with label set I then under suitable hypotheses the flag h vector of ∆ satisfies the linear relations hJ = hI\J , for J ⊆ I. The former are called the Dehn-Sommerville equations and the latter the generalized Dehn-Sommerville equations. From now on, we will just write them as DS for short. For more information see [32, Section 3.14] and the references therein. The DS equations are a form of Alexander duality. Our goal for this section is to formulate an axiom relevant to this notion of duality. This will also give a local PROJECTIONS, SHELLINGS AND DUALITY 25 version of the DS equations. As motivation we begin with a result that introduces the concept of shelling reversal. Theorem 4. Let ∆ be a shellable complex such that every facet lies in at least 2 chambers. Then ∆ is thin ⇒ |{D | R(D) = D}| = 1 ⇔ ∆ is homotopy equivalent to a sphere. If ∆ is thin then it satisfies the DS equations. Proof. The first part is proved in [9, Theorem 1.5]. We prove the second part. Also we assume that ∆ is labeled. If it is not then the same proof gives us the DS equations instead of the generalized DS equations. The proof is based on the idea of shelling reversal. Let ≤S be a shelling order of ∆, and let ≤ S be the reverse order defined by E ≤ S D ⇔ D ≤S E. Assume for the moment that ≤ S also gives a shelling of ∆. Now if we let R (resp. R ) be the restriction map of ≤S (resp. ≤ S ) then it follows that type of R(D)= I \ type of R (D).

~~This along with the fact that hJ = |{D | R(D) has type J}| gives us the generalized DS equations and proves the second part.~~

~~D G1 G2~~

~~F E~~

~~′ Figure 11. An example with FD = {G1, G2} and FD = {G3}.~~

~~Hence we only need to show that ≤ S gives a shelling of ∆. We assumed that ≤S is a shelling of ∆. This says that D ∩ (∪E~~

Claim: D ∩ (∪D< EE)= ∪G∈F ′ G. S D Proof of claim. (⊇) This is clear. (⊆) Let F be a face of D such that F * RHS. We show that F * LHS. Let E ≥ F such that E 6= D. We need to show that E

Remark. For a sketch of the same argument in the context of polytopes, see [34, pg 252]. Also for the first part of the theorem, we do not know whether thinness is equivalent to the other two conditions. The opposite axiom. Let ∆ be a thin shellable complex. Let C and D be the first and last elements of a shelling ≤S of ∆. Then C and D are the unique elements satisfying R(C)= ∅ and R(D)= D, where R is the restriction map of ≤S. As we saw in the proof of Theorem 4, the shelling ≤S is reversible and for the reverse shelling ≤ S , the roles of C and D get interchanged. Hence we would like to think of C and D as chambers opposite to each other. Motivated by this discussion, we add an opposite axiom to the axiomatic setup of Section 4. Of course, the main case of interest is when ∆ is thin. We give three equivalent definitions which correspond to the projection, restriction and shelling cases respectively. Their equivalence will be proved in Theorem 5.

~~G C D = GC E = GC C R (D) C R (D) C~~

~~Figure 12. How the opposition map works.~~

Let − : C→C be a map such that (P4) For any facet G of ∆ and any C ∈C, we have GC 6= GC. (R4) For all C,D ∈C, we have type of RC (D)= I \ type of RC (D). (S4) For all C ∈C, the partial orders ≤C and ≤C are dual to each other. If ∆ is not labeled then axiom (R4) is to be interpreted as saying that RC (D) and RC (D) are complementary faces of D. If ∆ is thin then axiom (P 4) says that for a facet G, GC and GC are the two distinct chambers containing G. Each of these axions implies that − : C→C is an involution of ∆. Also, using these axioms, C, “the chamber opposite to C”, can be characterized as follows in the three cases. P. C is the unique chamber such that FC 6= C for any F < C. R. C is the unique chamber satisfying RC (C)= C. S. C is the unique maximal element in the partial order ≤C . We now give a local version of the DS equations for a complex that satisﬁes the opposite axiom.

~~Lemma 6. If ∆ satisﬁes the opposite axiom then hJ (D)= hI\J (D).~~

Proof. Recall that hJ (D) = |{C | RC (D) has type J}|. Now by axiom (R4), the chamber C contributes to hJ (D) if and only if its opposite chamber C contributes to hI\J (D). Averaging the above local equations over all D ∈C leads to the usual generalized DS equations. (Of course, we already knew from Theorem 4 that ∆ satisﬁed the generalized DS equations.) In the unlabeled case, the same reasoning gives a local version of the DS equations instead of the generalized DS equations. PROJECTIONS, SHELLINGS AND DUALITY 27

Theorem 5. Let ∆ be a thin chamber complex that satisfies our (previous) axioms. Then ∆ satisfies (P 4) ⇔ ∆ satisfies (R4) ⇔ ∆ satisfies (S4). Proof. The complex ∆ is shellable since it satisfies our earlier axioms. And a shellable complex is automatically gallery connected. So really the only additional assumption on ∆ is that of thinness. To go from one set of axioms to another, we again use the formal connections outlined in Section 4.2. (P 4) ⇒ (R4). Let D ∈C. Apply axiom (P 4) to every facet of D. By the definition of RC and RC , we get that the type of RC (D)= I \ type of RC (D). (R4) ⇒ (S4). For a thin shellable complex, Lemma 2 gives a simpler description of ≤C . It is the transitive closure of the relation: E ≤C D if E is adjacent to D and RC (E) ≤ D. Let D and E be adjacent chambers and D 6= E. Let G be the common facet. By thinness of ∆ and axiom (R2), either RC (D) ≤ G or RC (E) ≤ G. This implies that either RC (D) ≤ E or RC (E) ≤ D. In order to prove axiom (S4), we need to show that D ≤C E ⇔ E ≤C D. We show that D ≤C E ⇒ E ≤C D. The other implication follows by symmetry. Suppose that D and E are adjacent and RC (D) ≤ E. Then by axiom (R4), we have RC (D) 6≤ E. Hence RC (E) ≤ D which implies E ≤C D. (S4) ⇒ (P 4). Let G be any facet of ∆. Since ∆ is thin, there are exactly two chambers in ∆≥G. Axiom (S1) says that for any C ∈ C, the partial order ≤C restricted to C≥G has a minimum. So the two chambers in ∆≥G get ordered by ≤C . Now axiom (S4) says that the order obtained using ≤C is reverse of the one got using ≤C and hence axiom (P 4) follows by the definition of projection maps. A simplicial hyperplane arrangement clearly satisfies axiom (P 4). Axiom (R4) can also be checked directly using the discussion in Section 7.4. We record this formally as a simple corollary. Corollary. A simplicial hyperplane arrangement satisfies all three opposite axioms (P 4), (R4) and (S4).

10. Duality in Buildings In the previous section, we axiomatized duality in a way that was relevant to thin chamber complexes. Now we prove a somewhat isolated result on duality in buildings and illustrate it with the building of type An−1. A suitable generalization of the opposite axiom to buildings may be possible. An appropriate setting may be gated chamber complexes of spherical type [21]. Also the duality result on buildings (Theorem 6) may generalize to Moufang complexes [22, Section 5]. We ﬁrst set up some notation. Let ∆ be a ﬁnite (spherical) building and A be the complete set of apartments of ∆. For chambers C,D, we denote by AC (resp. AC,D) the set of apartments in A that contain C (resp. C and D). Also, for a chamber C, let Cop denote the set of all chambers that are opposite to C. Observe op that there is a bijection between the sets C and AC as follows. We associate to op C ∈ C , the apartment Σ ∈ AC , which is the convex hull of C and C. Now let ′ op AC,D = {E ∈ C | D lies in the apartment determined by C and E}. Observe op that the bijection between AC and C restricts to a bijection between the sets ′ AC,D and AC,D. Also we let ρΣ,C : ∆ → Σ denote the usual retraction. 28 SWAPNEEL MAHAJAN

~~Lemma 7. Let C and C be a pair of opposite chambers in ∆ and let D be any chamber in the apartment determined by C and C. Then |AC,D||AD,C | = |AD|.~~

~~w u D w u C u w C~~

~~E E~~

~~Figure 13. A schematic picture.~~

Proof. Let δ : C × C → W be the W -valued distance function on C, where W is the Coxeter group of ∆. Since ∆ is spherical the Coxeter group W is ﬁnite. Denote the longest word in W by w0. Let C,D and C be as in the statement of the lemma. Also let δ(C,D)= u and δ(D, C)= w. Then uw = w0 since C and C are opposite ′ and C,D and C lie in the same apartment. Note that AC,D = {E | δ(D,E)= w}. ′ ′ This is because l(uw)= l(u)+l(w). We now deﬁne a bijection AC,D ×AD,C →AD ′ ′ as follows. If E ∈ AC,D and E ∈ AD,C then δ(E,E) = uw = w0 again because l(uw)= l(u)+ l(w). Hence E and E are opposite. So we map (E, E) to the unique apartment containing E and E. The map is clearly a bijection.

Let ∆ be a finite irreducible Moufang building. By irreducible, we mean that the Coxeter group W associated to ∆ is irreducible. We will not state the Moufang condition here; see [26, Chapter 6]. We only mention that all buildings of rank greater or equal to 3 are Moufang. Hence it is a restriction only when rk(∆) = 2. By a theorem of Tits [33, Theorem 11.4], if ∆ is irreducible and Moufang then it is the building of an algebraic group G over a finite field Fq. The group G(Fq) acts by simplicial type-preserving automorphisms on ∆. Also the action is transitive on pairs (Σ,C) consisting of an apartment Σ ∈A and a chamber C ∈ Σ. In particular, the stabilizer of a fixed chamber C, which we denote Stab(C) acts transitively on AC . It follows from the properties of buildings that for g ∈ Stab(C), ρΣ,C (gD) = ρΣ,C (D) for any chamber D ∈C. It is known that for any chamber C of ∆, the number of chambers adjacent to C along a fixed facet is a power of q. Here q is the cardinality of the finite field Fq. It follows that for chambers C,D, the number |AD| (and also |AC,D|) is a power of q. Using this fact, hJ (∆), the flag h vector of ∆, defined in Section 7 can be written as a polynomial in q. In the course of proving Theorem 6, we will recall the definition of hJ (∆) and give a precise definition of this polynomial. To emphasize the dependence on q, we write hJ (q) instead of hJ (∆). It will also follow that if Σ is any apartment of ∆ then hJ (Σ) = hJ (q) |q=1. The geometric significance of hJ (∆) was explained in Propositions 3 and 4. It counts the number of spheres in the homotopy type of ∆J , the type selected subcomplex of ∆. Now we prove the main result of this section. It gives the precise relation between the homotopy types of ∆J and ∆I\J . Theorem 6. Let ∆ be a finite, irreducible and Moufang building. Then with the −1 notation as above hJ (q)= hI (q)hI\J (q ). PROJECTIONS, SHELLINGS AND DUALITY 29

Proof. We start with a philosophical comment. “As q → 1, the building ∆ degen- erates to a single apartment Σ. The equation of the theorem then reduces to the generalized DS equation hJ (Σ) = hI\J (Σ).” Hence to prove the theorem, we will go in the opposite direction. We will start with the relation hJ (Σ) = hI\J (Σ). Then for a fixed pair of opposite chambers C, C ∈ Σ, we will use the retractions ρΣ,C and ρΣ,C to generalize this relation to ∆. We now start with the formal proof. We fix an apartment Σ and a chamber C that lies in it. Recall that RC (D) is the smallest face F of D such that FC = D (Section 4.2). Now set HJ (∆) = {D ∈ ∆ | RC (D) has type J}. Hence by definition, the cardinality of this set is hJ (∆). Similarly let HJ (Σ,C) = {D ∈ Σ | RC (D) has type J}. We make a few observations. The restriction map RC for ∆ when restricted to the apartment Σ gives the restriction map for Σ. This follows, for example, from the fact that the projection maps for ∆ are defined using the projection maps for the apartments. Hence the cardinality of HJ (Σ,C) is hJ (Σ). Next note that Σ is a Coxeter complex and hence corresponds to a simplicial hyperplane arrangement. So it satisfies the opposite axiom (R4); see the Corollary to Theorem 5 and the preceding comments. Hence applying axiom (R4) to it, we get HJ (Σ,C) = HI\J (Σ, C). This is a more precise set theoretic version of the generalized DS equation. The cardinalities of the two sets are hJ (Σ) and hI\J (Σ) respectively and do not depend on C and C. We next claim that −1 HJ (∆) = ⊔D∈HJ (Σ,C)ρΣ,C (D).

To see this, note that ρΣ,C (F )C = ρΣ,C (FC) for any face F of ∆. Hence F = RC (D) ⇔ ρΣ,C (F )= RC (ρΣ,C (D)) which proves the claim. Let C be the chamber −1 op opposite to C in Σ. Then ρΣ,C (C)= C , the set of chambers opposite to C. Also note that hI (Σ) = |HI (Σ,C)| = 1with C as the only element of HI (Σ,C). Applying the claim for J = I, we get hI (∆) = the number of spheres in the homotopy type op of ∆= |C | = |AC |. We now have the following identities.

−1 |AD| hJ (q)= |ρΣ,C (D)| = . |AC,D| D∈HXJ (Σ,C) D∈HXJ (Σ,C) |A | h (q)= |ρ−1 (D)| = D . I\J Σ,C |AD,C | D∈HXI\J (Σ,C) D∈HXJ (Σ,C) The ﬁrst equality is clear. For the second equality we apply Tits theorem to get a group G(Fq) associated to ∆. The group serves two purposes. It gives us the number q and it acts nicely on ∆. As already noted Stab(C) (resp. Stab(C)) acts transitively on AC (resp. AC ) and is compatible with the retraction ρΣ,C (resp. ρΣ,C ). This along with the fact |AC | = |AD| = |AC | gives us the second equality. The formulas show that hJ (q) can be written as a polynomial in q as claimed earlier. Furthermore due to the group action, it does not depend on the choice of Σ and C. It is also clear that if Σ is any apartment of ∆ then hJ (Σ) = hJ (q) |q=1. Recall that hI (q) = the number of spheres in the homotopy type of ∆ = |AD|. −1 Now if we replace q by q in the second equation and multiply by hI (q), we get

~~−1 |AD,C | hI (q)hI\J (q )= |AD| = |AD,C |. |AD| D∈HXJ (Σ,C) D∈HXJ (Σ,C) The theorem now follows from Lemma 7. 30 SWAPNEEL MAHAJAN~~

n Example 5. Building of type An−1: Let V be the n-dimensional vector space Fq , where Fq is the field with q elements. Let Ln be the lattice of subspaces of V under inclusion. Also let Bn be the Boolean lattice. Note that the flag (order) complex ∆(Bn) is the Coxeter complex of type An−1 and corresponds to the braid arrangement. Also note that a choice of a basis for V gives an embedding of Bn into Ln. The building of type An−1 over Fq is defined simply as the flag (order) complex ∆(Ln). The subcomplexes ∆(Bn), for various embeddings Bn ֒→Ln, play the role of apartments. The algebraic group G(Fq) in this case can be taken to be GLn(Fq) or SLn(Fq). n (2) For any chamber D, one may directly check that |AD| = hI (q) = q . Now fix an apartment Σ corresponding to a basis e1,e2,...,en of V . Let C = [e1] < [e1,e2] < ... < [e1,...,en]. Then the chamber opposite to C in Σ, namely C =

~~[en] < [en,en−1] <...< [en,...,e1]. If we let D = [ei1 ] < [ei1 ,ei2 ] < [ei1 ,...,ein ] be any chamber in Σ then a simple counting argument shows that~~

~~|{(j,k)|j~~

~~|{(j,k)|jik}| |AD,C | = q .~~

Hence we can see directly that |AC,D||AD,C | = |AD|. This proves Lemma 7 for this example. To unwind Theorem 6, we need to understand how retractions work. We leave that out and instead just give some explicit computations that illustrate the 2 2 3 theorem. For type A3, h∅ = 1, h{1} = h{3} = q(1+q+q ), h{2} = q(1+2q+q +q ), 3 2 2 2 3 6 h{1,2} = h{2,3} = q (1 + q + q ), h{1,3} = q (1 + q + 2q + q ), h{1,2,3} = q . We mention that these polynomials can also be described using descent sets without any reference to retractions, see [32, Theorem 3.12.3].

~~11. Future prospects In this paper we presented a theory of projection maps and compatible shellings. We now suggest some problems for future study.~~

The space of all shellings. As we saw, compatible shellings on a complex ∆ consist of compatible partial orders ≤C , one for every chamber C of ∆. Using this information, we may construct a geometric object S(∆) containing a lot of shelling information about ∆. It would be worthwhile to make this idea precise since the object S(∆) would be important for studying the space of all shellings of ∆. In the example of Section 4.3, a candidate for S(∆) is a line with vertices at the integer points. The group Z acts on S(∆) by translations. The complex ∆ is a quotient of S(∆) by the subgroup nZ of Z whose generator shifts a vertex n units to the right. In this regard, the following formalism could be useful. Let ∆ be a pure simplicial complex that satisﬁes our axioms. Let S be the space of all partial orders on C that extend ≤C for some C ∈C. Then S is a poset with the order relation given by extension with the partial orders ≤C as the minimal elements. We also think of C as a poset with the trivial partial order. Then S and C are posets related by a Galois connection. Denote a typical element of S by ≤S. We are slightly abusing notation because in our earlier usage, ≤S was always a linear order. The map S→C sends ≤S to the chamber that appears ﬁrst in ≤S. And the map C→S sends C to ≤C . PROJECTIONS, SHELLINGS AND DUALITY 31

Other questions. In contrast to the situation in Section 6, a shelling of a simplicial complex ∆ may have nothing to do with its gallery metric; see the example of Section 4.3. We ask for other non-metrical examples of ∆ that satisfy our axioms. While we explained some tiny applications and connections of our theory to the flag h vector, random walks and buildings, other applications of this circle of ideas still remain to be seen. As a possibility, we suggest that a general framework to study Solomon’s descent algebra [30] would be to start with a labeled simplicial complex ∆ that satisfies our axioms. If ∆ is a Coxeter complex then we would recover the usual descent algebra; see [4, 14, 20] for more information on this geometric way of thinking. As shown by the examples in Section 2 our theory is closely related to LRBs. We ask if this connection can be clarified. For example, we may ask the converse question to the discussion in Section 5. Given a simplicial complex that satisfies our axioms, how close is it to being a simplicial LRB (see the remark in Section 4.3)?

More examples. More examples of the theory that we presented may be possible. A possibility already mentioned is that of shellable complexes with group actions. Another possibility is that of simplicial polytopes equipped with a suitable class of shellings like line shellings. However a better alternative seems to be to generalize the theory itself. We conclude by suggesting some approaches through examples. (i) Recall that in the metric setup, FC was the chamber containing F closest to C. What happens if there is no closest chamber? We present a building-like example where this occurs.

~~FC?~~

~~FC? C~~

~~Figure 14. The Petersen graph - an almost-building?~~

3 1 v 4 The Petersen graph ∆: Consider a node v with 5 labeled leaves 2 5. This corresponds to the empty face of ∆. To get the rest of ∆, we do a “blowup” at v. The trees of the form (resp. ) are the vertices (resp. edges) of ∆. The 5 leaves of each tree are labeled 1, 2, 3, 4 and 5 in some order. Define an apartment to be the subcomplex made of those edges (chambers) that have the i j form for a fixed i,j. There are 10 apartments, 15 chambers and every chamber lies in exactly 4 apartments. Furthermore each apartment is an hexagon and given two chambers, there is always an apartment Σ containing them. So we may first define projection maps in Σ as usual and then try to extend them to ∆. However, in some cases, there are two apartments containing a face F and a chamber C and the products are not always consistent. This happens exactly when F and C are opposite to each other in some pentagon as shown in Figure 14. Then there are exactly two apartments containing F and C one of which is indicated in 32 SWAPNEEL MAHAJAN the figure. And the two candidates for FC are indeed the two chambers containing F that are closest to C. (ii)In[8], Björner constructed many examples of shellable complexes ∆. For example, he shows that all finite semimodular and supersolvable lattices are shellable. As a very special example, we would also like to mention the Whitehead poset which has found an application to group theory [12]. In all these examples, it is clear that ∆ has many shellings. It would be worthwhile to understand the precise compatibility relations among these shellings. (iii) New examples from old: Let ∆ be a labeled simplicial complex that satisfies our axioms. Then what can be said about its type selected subcomplex ∆J or its k- skeleton ∆k (if ∆ is not labeled)? We may ask the same question for the barycentric subdivision ∆′. These complexes again have many shellings but they do not directly give us new examples. However they might point to an appropriate generalization of the theory. (iv) Finally, we mention non-simplicial examples like polytopes and hyperplane arrangements. The second situation may be easier to handle since we already have projection maps in that case. The other non-simplicial examples to consider are LRBs as explained in Section 5. We mention that there is a notion of shelling for a poset. It is called a recursive coatom ordering [10]. This would be the relevant shelling concept in this generality. Acknowledgments. I would like to thank my advisor Ken Brown for his en- couragement, sharing of his insights and patient reading of earlier versions of this paper. I would also like to thank H. Abels, P. Abramenko, M. Aguiar, L. Billera, A. Björner, R. Forman, J. Meier, B. Mühlherr and G. Ziegler for useful discussions, comments and correspondence.

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~~Department of Mathematics, Cornell University, Ithaca, NY 14853 E-mail address: [email protected]~~